XIX. Понятие тождества в математической логике (к стр. 81).

The desire to exclude from the sphere of science any talk about numerical identity, as expected, appears most clearly where the scientific method is generally most rigorous and precise, namely in modern mathematical logic or so-called logistics. Here, in the “calculus of classes,” the identity of individuals is defined quite strictly, and is defined precisely as their joint membership in any class to which each of them can generally belong, i.e., in other words, as the possibility for any combination of characteristics of one individual to find a corresponding and equal combination of characteristics of another. It is clear that here the numeric identity is replaced by a specific identity. The salience of this substitution is all the more definite because in mathematical logic an individual is strictly distinguished from a single or special class corresponding to it (a class with a volume of “one”) and that, in the “calculus of relations,” an important axiom is posited, according to which each pair of given individuals has a special relationship that does not exist between any other two individuals, i.e., a relationship exclusively peculiar to this pair. And yet, despite these subtle distinctions, the identity of individuals is entirely decomposed in modern science into a set of common characteristics, so that the real character of the individual, as the bearer of his characteristics, in contrast to his formal character, is again only affirmed, but not expressed at all. This proves once again that numerical identity can only be symbolically relied upon or asserted, but is not defined, formulated or expressed logically. After these general remarks, let us briefly remind ourselves how these definitions are expressed in the symbolic signs of logistics. As is known, what is most striking here is the decisive distinction between the relations of “implication” and “inclusion,” i.e., the inclusion of a judgment or class in another judgment or class, from the operation of subordinating an individual to a class or, accordingly, to a judgment. Ɔ is a sign of both implication and inclusion; ε - relative to the class - the sign of the operation of establishing correspondence (abbreviation ἐστί), and z - the sign of the same operation regarding the judgment. This distinction, reinforced by the difference of signs, is extremely important. However, ordinary speech confuses both types of relations, i.e., implication with inclusion and the operation of correspondence, under the general designation of the connective “is”, “essence”; logicians identified them for a long time, and only Peano first recorded them, and even then thanks to the symbolism he invented. To make the difference between the operation Ɔ and the operation of correspondence more clear, let us take, for example, an ordinary syllogism: major: Every man is mortal. conclusio: Therefore, Socrates is mortal. The connective of the major premise here will be Ɔ, as is usually thought, but the connective of the minor premise is not Ɔ at all, as almost the entire majority of logicians tend to think, but – ε. In fact, in the larger premise the relation of the classes “humanity” and “mortality” is established, and in the second it is no longer the relation of classes, but of the individual “Socrates” to the class “humanity”, to which “Socrates” belongs. So the major premise is an undoubted implication, as is customary to say, but the minor premise is no longer an implication, but the subordination of the individual to the class, and precisely of the first type. This means that the formula of the syllogism being analyzed will actually be: a Ɔ b . ∩ . x ε a: Ɔ: x ε b (I) and not at all the formula of an ordinary, typical syllogism establishing the relationship between classes: a Ɔ b . ∩ . c Ɔ a: Ɔ: c Ɔ b (II) It is hardly necessary to note the significant difference between formulas (I) and (II). One should not think that the symbols Ɔ and ε, having very different logical and ontological meanings, could be mixed with impunity from a formal point of view, for the purposes of numerical logistic mechanics, and that this “subtlety” of Pean’s distinction of concepts has no “pragmatic” significance for the technique of notation. Far from it, because the very properties of both relations, i.e. Ɔ and ε, are significantly different: the relation Ɔ, as establishing a connection between homogeneous entities or terms (classes, judgments) is transitive, while the relation ε, as establishing a connection between heterogeneous entities (class and individual) is obviously intransitive: if a Ɔ b and b Ɔ c, then it is clear that, according to the formula a Ɔ b . ∩ . b Ɔ c: Ɔ: a Ɔ c (III), and a Ɔ c; but from the fact that x ε y and y ε z, it does not at all follow that x ε z, for if x, as an individual, is subsumed under the class y, and the class y, as an individual, is subsumed under the class z, then z will already be in relation to x a class of classes, and not just a class, and therefore cannot count x among its elements, as part of its volume; for z, the volume consists of individual classes, and they themselves, for z, are already indivisible, indivisible, indecomposable. With inattention to the intransitivity of the relation ε, which in turn is based on the confusion of ε with Ɔ, sophisms are often constructed, the formal and responsive exposure of which is not always easy. Such, for example, are some of the sophisms in Plato’s “philosophical comedy” Euthydemus 850, well, at least reasoning like the following: “Gold is gold and cannot be not gold; a person is a person and cannot be not a person. Consequently, your father Kheredem, - approximately this is what Socrates says to Euthydemus, - is a father and he cannot be a non-father. This means that he is the father of everyone, and not only to people, but also to horses and other animals. In the same way, your mother is the mother of everyone, the mother of hedgehogs. So you are a brother to calves, and puppies, and piglets. “Then, you have a dog, and he has puppies and, therefore, the dog is their father. But the dog is yours. And he is yours, being his father, so your father is a dog, and you are a brother to the puppies. “Next: you beat your dog, that means you beat your father, etc.” Or, here’s another example from “Hippias the Greater” 851: “Each of us, the two interlocutors, is one and, therefore, it is common for each of us to be an odd number. Consequently, both of us together will also be an odd number when there are two of us. But if this is not so, if both of us together are even, then each of us separately is also even, etc.” The reasoning is carried out here according to the following scheme: Heredem (individual) ε your parent (class); your parent (individual) ε parent (class); parent(class) = parent(class); producing hedgehogs, piglets, etc. (class) Ɔ Ɔ birthing creatures (class). But it is clear that, due to the intransitivity of the operation “ε”, from these premises it is in no way possible to draw the conclusion that Heredem ε producing hedgehogs, etc. In the same way, from the fact that Peter (individual) ε apostle (class of apostleship), i.e., that Peter is also “twelve”, and not “one 852. So, from what has been said, it becomes absolutely certain that even from a purely formal point of view, the individual is fundamentally different from the class, even from the individual class, contrary to the opinion of nominalist logicians, who sought to interpret the class not as a single and self-contained body of thought, but as a collection of individuals, and contrary to the aspirations of positivist logicians, who wanted to destroy the original nature of the individual and reduce him to the sum of characteristics, i.e. to a unit class. A “special” or “single” - singulière - class must be strictly distinguished from the only individual included in its volume: otherwise one could write for such a class x the formula: which, as we have already seen, is meaningless, for “ε” is a sign of a relationship between different and even heterogeneous terms, and not one and the same. A unit class formed from a single element “x” is therefore usually denoted by a special sign, namely ιx, read: “equal to x”, “égal à x”; here the symbol “ι” is an abbreviation of the word ἴσος, equal. This symbol "ι" is formally defined by the equality: i.e. “ιx is a symbol of a class y such that (ɜ) justifies the propositional (continuous) function (with variable y) “y = y.” Hence, applying to both sides of the written equality the operation “уɛ”, which brings the individual element “y” equal to the classes “ɩx” and “уɜ (у = x)”, and remembering that the operations “yɛ” and “уɜ” mutually destroy each other, we find: y ε ι x . = . y = x(VI), i.e. the equality of two symbols “ει” and “=”, so that It follows that, although formula (IV) is unfair, however (for x = x), i.e., that the individual “x” always belongs to his unit class “ɩx”. If, on the contrary, “a” is a unit class, then its only element can no longer be designated through “a”, but must be used by special symbols, which include the inverted symbol “ ”, which reads: “this a”, “le a”, “der a”, “ὁ a”. In general, the symbol "ɩ" transforms an individual into its unit class, and vice versa, the symbol " » transforms the unit class into an individual, so we have two equalities. equivalent to each other; in signs: Everything that has been said so far is completely fair, because it is based on the essential difference between the individual class and the individual. But the individual is introduced here only by the symbol, without definition. Therefore, for rationalism there is an obvious stumbling block here. They try to get around it in the following way: A unit class is defined as a class formed by a single individual. But what is the number one? And what is an “individual”? Mathematical logic, “according to the custom of the mathematical sciences – selon l’habitude des Mathématiques” 854 does not determine the individual, but only the identity of individuals. Of course, the point here is not at all in the imaginary “custom of mathematical sciences,” but in the impossibility of defining the individual as a super-rational reality. An attempt to determine the identity of individuals makes it possible to replace the question of real numerical identity with a question of attribute, formal similarity, i.e. reasoning - impossible! - over individuals - reasoning over concepts about them, i.e. over classes. This substitution is done deliberately, and it is deeply significant, especially after the decisive distinction between individuals and classes. So, “two individuals k and l are said to be identical if the second belongs to every class in which the first participates - on dira que les deux individus k et l sont identiques, si le second appartient à toute classe dont le premier fait partie” 855. Symbolically, this definition is expressed by the formula: k ≡ l: k ε a. Ɔa. l ε a (XI). Here “α ≡” is an identity sign, and the pointer “a” for Ɔ means that the written implication is valid for any a that satisfies the inclusion “k ε a” 856. At the same time, attention is drawn to the fact that “the identity of individuals is logically different from the equality of classes, just as individuals k and I are different from the individual classes ɩ k and ɩ l” 857. How can we understand this formula? – Nothing more than the definition of the sign “α ≡”. Formula (XI) says that when we encounter a hitherto unprecedented graphic combination of dashes and letters, a “picture” which, therefore, does not have any meaning, then, from now on, once and for all, we want, we believe, we demand to understand by it nothing more than an abbreviated, conventional designation of an implication that is already clear to us or, more precisely, a set of implications, with all possible values of the variable “a”, the field of change of which is determined by the function “k e a”, i.e., we repeat once again, by “k ≡ l” we want to mean an abbreviated designation for the expression where P is the sign of the logical multiplication of all factors obtained for all possible values ​​of a. It is this system of implication, which speaks only about the relationship between the belongings, - ε, - of individuals “k” and “l” to classes “a”, that we agree to call the identity of individuals. But we still do not know logically what an individual is, we do not have the concept of an individual and, therefore, we only posit the term purely symbolically, as a sign of something (- anything, but not a class or a judgment or a relationship -) that can be in a relation of “identity”; by identity we mean some complex formula in relation to this “something” to classes. We repeat once again that here, in the most obvious way, there arises the powerlessness of logical thought in the face of concrete, i.e., individual being, and the pitifulness ( - necessary pitifulness! - ) of the mind's attempts to replace individual being with rational-figurative - but not rational! - terms. – Further, as stated, the question of determining the uniqueness of the class remains open. How, in fact, can one determine, in rational terms, that the class ɩa is singular, that is, that it contains only one element - that there is only one a? – This is achieved by indicating two characteristics of the class ɩa: firstly, that it contains elements at all, i.e. that it is a non-zero class; secondly, that if there were two such elements, namely x and y, then they would be identical to each other. That the class ɩa is not zero is expressed by the negative formula: and “is not” (– =) Λ (XIV), where Λ is the sign of the zero class, or else, in a more convenient, positive form: So, the singularity of the class ɩa will be expressed through the formulas: a – = Λ: x ε a . y ε a. Ɔ x,y. x ≡ y (XVII), Ǝa. x ε a . y ε a . Ɔ x,y. x ≡ y (XVIII) 859 , that is, for any x and y belonging to the class ɩа, the written implication remains true. It is hardly necessary to point out that everything said about the logical definition of identity also applies to this definition of singularity, for singularity is only a special case of identity, namely identity with oneself, self-identity. Turning, finally, to the logic of relations - rélations - we here naturally must resolve the question of the relationship between individuals. In this department of mathematical logic it is taken as an axiom that there is always a relationship between individuals, which is itself individual, itself represented by a unique individual. “Between two given individuals,” says the axiom, “there exists a special—singulière—relation that does not exist between any two other individuals.” However, this special relationship is explained, as one would expect, in a formal sense, and not in a real sense. This is made clear from the explanations that follow this axiom. “From the point of view of volume - extension,” they say, “this axiom is obvious, because the pair in question is enough to determine a relation that is different from all others. From the point of view of content, we can say that if we consider the totality - l'ensemble - of all the relations that exist between two given individuals, then the same totality does not exist between any other pair of individuals; in other words, if a certain pair has all the relations of another pair, then these two pairs are identical, which is written like this: x1 R y 1 · ƆR · x 2 R y2 : Ɔ: x 1 ≡ x2 · y 1 ≡ y2 (XIX) 860 Here the sign ƆR shows that the complication of the left side is valid for any relation R that can connect x 1 and y 1 . So, individual, i.e. specifically, the relationship again decomposes into a number of general ones, i.e. abstract, - all abstract relations included in the composition of this particular one. But, in addition to the ontological senselessness of such equating the concrete with the sum of abstractions, the question arises about the legality of such a definition, even in the purely formal area. It, after all, is entirely based on the concept of “all” relationships between a given pair of elements. Not to mention that the very concept of “everything” is not yet defined in mathematical logic, especially when it relates to the group of super-finite groups, it is doubtful Is it possible, in general, to have a definite concept about the group of “all” relations not between classes, but between individuals. But this is what requires proof, because it is not at all clear (and it is hardly acceptable at all) that a transfinite group of abstractions can exhaust concreteness. If specificity can be considered from a formal-rational angle of vision, then it, undoubtedly, can be introduced into formal speculation only under the guise of a limit, i.e., as an absolute maximum. But the concept of such has not yet been developed, except for a completely unknown attempt by Archimandrite. Serapion - and besides, it is unclear whether such a thing can even exist in relation to individuals. After all, specific individuals possess creativity, are capable of creating absolute, unforeseen relationships that are not included in an arbitrarily voluminous group of ready-made relationships - in a word, they exceed any preconceived concept about them: in the words of A. Bergson, “la vie déborde l’intelligence - life emerges from the shores of reason,” and this always happens. Plat. , – Euthydemus (pp. Solovyov, vol. 2, p. 145). Pl. , – Hippius the Greater, id., vol. 2, pp. 133–140. Cout., – Pr. d. M., pp. 51–52.